。
I managed to talk her into going on a date.
Literal
I [topic-は] her [object-を] skillfully persuaded-[sequential-て] date-did.
うまく is the adverbial form of うまい ('skillful, good at'), meaning 'skillfully, successfully.' 説得する ('to persuade'). デートする ('go on a date,' with デート as an English loanword). The ~て sequential links persuasion and dating.